Modeling
is of primary importance for contemporary engineering design process on
both device and system levels. Unfortunately, available
high-fidelity models are typically based on computationally expensive
computer simulations (e.g., finite element analysis). High CPU cost is
undesirable from the point of view performing tasks such as parametric
design optimization, statistical
analysis and yield optimization. Also, sensitivity information is
normally unavailable and the high-fidelity models are often
analytically intractable (e.g., non-differentiable, discontinuous)
which excludes using traditional, gradient-based methods for the design
optimization.

Surrogate
modeling [1] is a way of replacing the original, computationally
expensive and otherwise unmanageable model, by a computationally fast
surrogate for the purpose of performing the design tasks. Typically, the surrogate
model is established locally, in the limited region of interest and, to
ensure sufficient accuracy, it is based on some amount of data from the
original, high-fidelity model.Functional versus Physical Models. Space MappingThere
is a variety of surrogate modeling techniques that can be categorized
into functional and physical ones. Functional surrogate models are
based on appropriate function approximation/interpolation of the
sampled high-fidelity (fine) model data. Popular approaches include
polynomial approximation, radial basis functions [2], kriging [3] and
neural networks [4].

Physical
surrogates, on the other
hand, exploit physically-based low-fidelity models (e.g.,
coarse-mesh simulations or analytical formulas). The surrogate is
typically built by enhancing the low-fidelity (coarse) model
by appropriate corrections terms derived from a limited amount of
the high-fidelity model data. Due to the fact that the low-fidelity
model encodes certain knowledge about the original
structure, physical surrogates typically exhibit better
generalization
properties than the functional ones.

Space mapping (SM) [5],
[6] is
a notable example of the physical surrogate modeling approach. Space
mapping enhances the low-fidelity model by composing it with simple
(usually linear) transformations with the parameters of these
transformations extracted to minimize misalignment between the
surrogate and the high-fidelity models at certain (small) number of
base points (designs). Formulation of the standard space mapping
modeling methodology as well as the exposition of recent advances
in SM modeling can be found in Koziel et al. 2008.

Space Mapping Modeling for Microwave Engineering

In
microwave engineering, high-fidelity models are normally based on
full-wave electromagnetic simulations, whereas low-fidelity models may
be using coarse-mesh simulations, equivalent circuits or analytical
formulas.

Illustration:
Surrogate modeling of the microstrip bandpass filter [7]. The
high-fidelity model is implemented in the electromagnetic simulator FEKO:

The low-fidelity model is a circuit equivalent model implemented in Agilent ADS:

The
surrogate model is set up using input and implicit space mapping [5]
using a star-distribution base set [6] (here, 11 high-fidelity model
simulations). Responses of the high-fidelity (solid line) and the
low-fidelity (dashed line) models at the reference design:

Responses of the high-fidelity (solid line) and the space mapping surrogate (dashed line) models at the reference design:

Application
2: Yield estimation. Surrogate model was used to estimate the yield
assuming 1% deviation for all design variables. Estimation performed
for 200 random samples. Estimated yield is 62% (left picture). Similar
estimation performed directly on the high-fidelity model gives 52%
(right picture):

EOMC Focus

The techniques developed in EOMC are mostly based on the
physical surrogate modeling principle, particularly space mapping as
well as combination of space mapping with various function
approximation methods.

EOMC focuses on computationally
efficient surrogate modeling methods for microwave/RF engineering and
their applications for statistical analysis and design optimization of
microwave devices and circuits. On the top of standard
space mapping, EOMS is exploiting other techniques such as
adaptive response correction [8] and shape-preserving response
prediction [9]. EOMC works on the development
of user-friendly software implementing surrogate modeling
techniques with special emphasis on interfacing commercial EM/circuit
simulators.References